A-numerical radius inequalities for semi-Hilbertian space operators

Let A be a positive bounded operator on a Hilbert space (H, <., .>). The semi-inner product < x, y >(A) := < Ax, y >, x, y is an element of H induces a semi-norm parallel to . parallel to(A) on H. Let parallel to T parallel to(A) and w(A)(T) denote the A-operator semi-norm and the A-numerical radius of an operator T in semi-Hilbertian space (H, parallel to . parallel to(A)), respectively. In this paper, we prove the following characterization of w(A)(T) w(A)(T) = sup(alpha 2+beta 2=1) parallel to alpha T+T-#A/2 + beta T - T-#A/2i parallel to(A), where T-#A is a distinguished A-adjoint operator of T. We then apply it to find upper and lower bounds for w(A)(T). In particular, we show that 1/2 parallel to T parallel to(A) <= max {root 1 - vertical bar cos vertical bar T-2(A), root 2/2}w(A)(T) <= w(A)(T), where vertical bar cos vertical bar T-A denotes the A-cosine of angle of T. Some upper bounds for the A-numerical radius of commutators, anticommutators, and products of semi-Hilbertian space operators are also given. (C) 2019 Elsevier Inc. All rights reserved.